Proceedings of the

Annual Stability Conference

Structural Stability Research Council

Grapevine, Texas, April 18-21, 2012

On the Behavior, Failure and DSM Design of Thin-Walled Steel Frames

C. Basaglia1 and D. Camotim

1

Abstract

This paper reports the available results of an ongoing numerical investigation on the buckling, post-

buckling, collapse and design of simple frames. The results presented and discussed are obtained through

analyses based on Generalized Beam Theory (elastic buckling analyses) and shell finite element models

(elastic and elastic-plastic post-buckling analyses). Moreover, the ultimate loads obtained are used to

establish preliminary guidelines concerning the design of steel frames failing in modes that combine

local, distortional and global features. An approach based on the existing Direct Strength Method (DSM)

strength equations is followed and the comparison between the numerical and predicted ultimate loads

makes it possible to draw some interesting conclusions concerning the issues that must be addressed

by a DSM design procedure that can be successfully applied to thin-walled steel frames. 1. Introduction

The extensive use of thin-walled steel frames in the construction industry stems mostly from their high

structural efficiency (large strength-to-weight ratio), remarkable fabrication versatility and very low

production and erection costs. However, since these steel framed structures are usually built from open-

section members (e.g., cold-formed columns and beams), which are highly prone to local, distortional and

global buckling phenomena, the direct assessment of their structural response constitutes a rather complex

task (e.g., Dubina 2008). Therefore, such steel frames are currently designed by means of an indirect

approach, basically consisting of (i) “extracting” the various members from the frame (more or less

adequately) and then (ii) safety checking them individually as “isolated members”. The main shortcoming

(source of error/approximation) of this approach stems from the inadequate accounting of the “real

behavior” of the frame joints − indeed, the “extracted” members are almost always safety checked under the assumption of standard support conditions (pinned or fixed end sections) and the only “link” to the

original frame is the member “effective/buckling length”, a concept initially devised in the context of the

in-plane flexural buckling of isolated members and later extended to handle the geometrically non-linear

in-plane frame behavior. In particular, no attention is paid to several important frame joint behavioral

features, such as those associated with the (i) warping torsion transmission (e.g., Basaglia et al. 2012), (ii)

localized displacement restraints due to bracing systems or connecting devices (e.g., Camotim et al 2008)

or (iii) local/global displacement compatibility (e.g., Camotim et al. 2010). In order to overcome the

above shortcoming, Part 1-1 of Eurocode 3 (EC3-1-1 − CEN 2005) proposes (allows for) the use of a

1 ICIST, Instituto Superior Técnico, Technical University of Lisbon, Portugal.

,

so-called “General Method”, which is intended for the design a wide variety of structural systems, even if

no proper validation results or application guidelines are provided − this explains why several European Community countries either completely forbid or severely restricted its application in their EC3 National

Annexes. Quite recently, Bijlaard et al. (2010) investigated the application of this method to the design of

plane frames against spatial global failure (i.e., collapse mechanisms involving lateral-torsional buckling). In the last few years, a fair amount of research work has been devoted to the development of efficient

design rules for isolated (single-span) thin-walled steel members, mostly subjected to uniform internal

force and moment diagrams. The most successful end product of this intense research activity is the

Direct Strength Method (DSM), which (i) has its roots in the work of Hancock et al. (1994), (ii) was

originally proposed by Schafer and Peköz (1998) and (iii) has been continuously improved since then

(e.g., Schafer 2008). Following the universal acceptance of the DSM approach to design cold-formed

steel members, it has already been included in the latest versions of the corresponding North American

(AISI 2007) and Australian/New Zealand (AS/NZS 2005) specifications. The aim of this work is to present and discuss the results of an ongoing numerical investigation on the

local, distortional and global buckling, post-buckling, collapse and DSM design of simple frames. The

numerical results presented were obtained through (i) Generalized Beam Theory (GBT) buckling analyses

and (ii) elastic and elastic-plastic shell finite element (SFE) post-buckling analyses. In particular, some

interesting conclusions are drawn on the features that must be incorporated in a DSM design procedure

for this type of thin-walled steel structural systems. 2. Numerical Investigation: Scope and Modeling Issues

Fig. 1 shows the dimensions of the plain channel, lipped channel and I-section exhibited by the frame

members dealt with in this work − all have elastic constants E=205 GPa (Young’s modulus) and υ=0.3 (Poisson’s ratio). Figs. 2 to 4 depict the main features of the deformation modes that are more relevant

for the GBT buckling analyses carried out throughout the paper (i.e., those with significant

contributions to the frame buckling mode shapes).

150

200

4.0

5

60

Lipped channel I-sectionPlain channel

200

100

3.5

100

1.5

(mm)

Figure 1: Plain channel, lipped channel and I-section dimensions.

1 2 3 5 6 4 7

9 8

Global Local

Figure 2: Main features of the most relevant plain channel deformation modes.

1 2 3 4 5 6 7 8 9

Global Distortional Local

Figure 3: Main features of the most relevant lipped channel deformation modes.

1 2 3 4 5 6 7

Global Local

Figure 4: Main features of the most relevant I-section deformation modes.

The buckling, post-buckling and ultimate strength results presented next concern the non-linear behaviors

of the frames shown in Figs. 5 to 7. The “L-shaped” plane frame depicted in Fig. 5 (termed LF-U) is

formed by two orthogonal short members exhibiting (i) identical plain channel cross-sections (see Fig. 1),

(ii) fixed end sections with warping prevented, and (iii) flange continuity at the joint (i.e., the two

members are connected with their flanges lying in the same plane) − the members (A and B) have the same length (LA=LB=70cm) and are unequally axially compressed (NA=Q and NB=0.8Q, where Q is the

load parameter) − naturally, this setting “forces” the collapse to occur in member A. The symmetric orthogonal portal frame displayed in Fig. 6 (termed PF-C) (i) is formed by three members with identical

lipped channel cross-sections (see Fig. 1), (ii) has fixed column bases and joints with flange continuity,

and (iii) is acted by four loads applied at the joints and causing only first-order member axial forces

(NA=NC=0.83Q and NB=Q). Finally, Fig. 7 shows a second “L-shaped” plane frame (termed LF-I), now

formed by two fairly long members (LA=400cm and LB=600cm) exhibiting (i) identical I cross-sections

(see Fig. 1), (ii) again fixed end sections with warping prevented, and (iii) a box-stiffened joint (web

continuity) − the frame is subjected to a vertical load Q applied at the beam mid-span cross-section centroid, causing essentially (i) bending in member B (beam) and (ii) bending and axial compression in

member A (beam-column). Note that the geometries of these three frames were chosen in order to ensure

buckling and failure modes involving all types of deformations (local, distortional and global).

L =70cm

member A

Amember B

B

L =70cm

Y

AXXZ

BXQ

_

_

_

0.8Q

Figure 5: “L-shaped” frame formed by plain channel members (LF-U): geometry, loading and support conditions.

L =50cm

member A

A

member B

B

L

=7

0cm

member C

Q0.83Q

0.83Q

Q

Figure 6: Symmetric portal frame (PF-C): geometry, loading and support conditions.

cL

=

400cm

BX

XA

bL =600cmQ b

L /2

Y

Z X

θ >0

column

beam

BX

XA

column

beam

Figure 7: “L-shaped” frame formed by I-section members (LF-I): geometry, loading and support conditions.

Concerning the GBT and SFE analyses, the following modeling issues deserve to be mentioned:

(i) GBT Discretization. The GBT buckling equilibrium equations were solved using the beam finite

element originally developed by Bebiano et al. (2007) and subsequently enhanced by Basaglia et al.

(2010) − it is worth noting that the formulation of this element takes into account the geometrical effects stemming from the longitudinal normal stress gradients and ensuing pre-buckling shear

stresses, thus enabling the appropriate capture of shear buckling effects.

(ii) SFE Discretization. The elastic and elastic-plastic SFE post-buckling analyses were performed in

the commercial code ANSYS (SAS 2009) and based on frame discretizations into fine meshes

of 4-node Shell181 finite elements.

(iii) Material Modeling. The steel material behavior is deemed either linear elastic (bucking analyses) or

linear-elastic/perfectly-plastic with the von Mises yield criterion and its associated flow rule (post-

buckling analyses). This means that (iii1) no strain hardening is taken into account and (iii2) it is

assumed that there exists enough ductility to allow for all the stress redistributions that occurs

prior to the frame collapse.

(iv) Initial Imperfections. Critical-mode initial geometrical imperfections with amplitude equal to

either 10% of the wall thickness (local or distortional buckling) or L/1000 (global buckling),

values that are often adopted in numerical simulations carried out by thin-walled steel researchers

− a more judicious choice requires the performance of imperfection-sensitivity studies, which falls outside the scope of this paper. Moreover, no residual stresses are included in the analyses.

3. Buckling Results

In all existing design procedures, a crucial step consists of identifying the buckling mode nature, a task

by no means straightforward in thin-walled frames. This can be confirmed by examining Figs. 8 to 10,

which provide two representations of critical buckling mode shapes of the three frames considered in this

work, namely (i) ANSYS 3D views and (ii) GBT modal amplitude functions. The corresponding frame

critical buckling loads, yielded by GBT and ANSYS analyses, are the following: (i) Qcr.GBT=254.83kN

and Qcr.ANSYS=250.41kN, for the LF-U frame, (ii) Qcr.GBT=51.66kN and Qcr.ANSYS=51.37kN, for the

PF-C frame, and (iii) Qcr.GBT=51.32kN and Qcr.ANSYS=51.83kN, for the LF-I frame. The analysis of these

frame buckling results prompts the following remarks:

(i) The GBT and ANSYS critical buckling loads practically coincide − the maximum difference is 1.7% and concerns the LF-U frame. There is also very close agreement between the critical buckling mode

representations provided by the two analyses.

(ii) While the LF-U frame buckles in a pure local mode, the SF-C and LF-I frame critical buckling

modes are “mixed”, in the sense that they combine two types of deformation modes: (ii1) local and

distortional (PF-C), or (ii2) local and global (LF-I).

(iii) The LF-U and PF-C frames can be more accurately described as sets of rigidly connected columns,

since, in practical terms, all their members are axially compressed. Thus, they exhibit a “column-

like” buckling behavior that is triggered by the column with the “worst” combination of axial load

and end support conditions: (iii1) member A in the LF-U case and (iii2) member B in the PF-C case.

(iv) On the other hand, the buckling behavior of the LF-I frame, which exhibits a “real frame behavior”

(member A is subjected to axial compression and bending − beam-column), is triggered by the lateral-torsional instability of the beam (member B).

(v) The critical buckling mode shapes of the LF-U and PF-C frames exhibit practically null joint

deformations (i.e., displacements of the point corresponding to the intersection of the converging

LA (cm)

-1.0

0.0

1.0

0 10 20 30 40 50 60 70

k

5

7× (20)

9× (20)

-1.0

0.0

1.0

0 10 20 30 40 50 60 70

LB (cm)

φk

5× (20)

Figure 8: − LF-U frame: ANSYS and GBT critical buckling mode representations.

-1.0

0.0

1.0

0 10 20 30 40 50 60 70

LA (cm)

9 × (10)

5

7 × (2)

-1.0

0.0

1.0

0 10 20 30 40 50

LB (cm)

9 × (10) 5

7 × (2)

Figure 9: − SF-C frame: ANSYS and GBT critical buckling mode representations.

-1.0

0.0

1.0

0 100 200 300 400

LA (cm)

φk

3

4 × (5)

-1.0

0.0

1.0

0 100 200 300 400 500 600

LB (cm)

k

3 4 × (5)

5 × (5) 6 × (5)

7 × (5)

Figure 10: − LF-I frame: ANSYS and GBT critical buckling mode representations. member centroidal axes). However, significant out-of-plane displacements, which stem from the

beam lateral-torsional buckling (recall that the frame joint is not restrained against out-of-plane

displacements), occur in the close vicinity of the LF-I frame joint.

In order to establish the frame critical buckling mode “dominant natures”, it is necessary to perform GBT

analyses that including only global, distortional or local deformation modes. Table 1 shows the ratios

between the “pure” global (Qb.e), distortional (Qb.d) and local (Qb.l) buckling loads and Qcr.GBT. Then, the

frame “dominant buckling mode nature”, given in the last column, reflects that the corresponding “pure”

buckling load is the closest to Qcr.GBT (i.e., corresponds to the the lowest of the three ratios).

Table 1: Relation between the Qb and Qcr load values.

Frame Qb.e / Qcr Qb.d / Qcr Qb.l / Qcr Dominant buckling

mode nature

LF-U 26.75 − 1.00 Local

PF-C 5.22 1.03 1.52 Distortional

LF-I 1.05 − 1.71 Global 4. Post-Buckling Results

This section addresses the SFE analysis of the elastic and elastic-plastic (for yield stress values equal to

fy=250, 450, 650MPa) post-buckling behaviors of the LF-U, PF-C and LF-I frames. The curves shown in

Figs. 11(a), 12(a) and 13(a) are the post-buckling equilibrium paths Q vs. v1, Q vs. v2 and Q vs. θ, where (i) the symbols , and identify the limit equilibrium states corresponding to the ultimate loads, (ii) v1 and v2 are the transverse displacements of points P1 (LF-U frame) and P2 (PF-C frame), selected in

order to provide a better frame post-buckling characterization (Figs. 11(a) and 12(a) show the location of

these points), and (iii) θ is the torsional rotation of the beam mid-span cross-section (LF-I frame). As for Figs. 11(b), 12 (b) and 13(b), they provide the failure modes and von Mises stress distributions of

three frames: (i) LF-U frame with fy=250MPa, (ii) PF-C frame with fy=450MPa and (iii) LF-I frame

with fy=650MPa − note that Fig. 12(b) also includes detailed information showing the onset of yielding. The observation of the post-buckling results shown in Figs. 11 to 13 leads to the following conclusions:

(i) Naturally, the amount of post-critical strength reserve increases (i1) with the yield stress (obviously)

and (i2) in the frame sequence as LF-I (global buckling), PF-C (distortional buckling), LF-U

(local buckling). The higher post-critical strength reserve, occurring for the LF-U frame with

fy=650MPa, corresponds to an ultimate-to-critical load ratio equal to 1.90.

0

100

200

300

400

500

600

0 2 4 6 8 10 12 14 16

Q (kN)

v1 (mm)

fy=250MPa

450MPa

650MPa

Elastic Qu (kN)

263.06

377.26

476.28

18

7.5

mmv1

P1

0 27.78 55.56 83.33 111.11 138.89 166.67 194.44 222.22 250

MPa

fy=250 MPa

(a) (b)

Figure 11: LF-U frame (a) post-buckling equilibrium paths and (b) deformed configuration plus von Mises stresses at collapse.

0

20

40

60

80

100

120

0.0 2.0 4.0 6.0 8.0 10.0

Q (kN)

v2 (mm)

fy=250MPa

450MPa

650MPa

Elastic

Qu (kN)

55.90

77.15

93.40

( I )

( II )

member B

P2v2

154.6mm

0 50 100 150 200 250 300 350 400 450

MPa

fy=450 MPa

( II ) ( I )

Plastic

strain

(a) (b)

Figure 12: PF-C frame (a) post-buckling equilibrium paths and (b) detail of the onset of yielding and deformed

configuration plus von Mises stresses at collapse.

0

10

20

30

40

50

60

0.0 0.1 0.2 0.3 0.4 0.5

Q (kN)

θ (rad)

fy=250MPa

450MPa

650MPa

Elastic

Qu (kN)

34.81

44.87

51.09

0 72.22 144.44 216.67 288.89 361.11 433.33 505.56 577.78 650

MPa fy=650 MPa

(a) (b)

Figure 13: LF-I frame (a) post-buckling equilibrium paths and (b) deformed configuration plus von Mises stresses at collapse.

(ii) Increasing the yield stress from 250MPa to 650MPa leads to ultimate load increases of 81%

(LF-U frames), 67% (PF-C frames) and 47% (LF-I frames).

(iii) The members responsible for the frame collapse are those triggering its instability, i.e., member A

(LF-U frames), member B (PF-C frames) and the beam (LF-I frames).

(iv) As shown in Fig. 12(b), the onset of yielding in the PF-C frame with fy=450MPa occurs for the

load Q=52.30kN (equilibrium state I) and plasticity first appears at the member lips and in the

vicinity of the frame joint. The collapse occurs considerably after, for Q=77.15kN (equilibrium state

II), and is precipitated by the full yielding of the member B central region.

(v) While the LF-U and PF-C frame failure mechanisms are very similar to the corresponding critical

buckling mode shapes (predominantly local and distortional collapses − see Figs. 11(b) and 12(b)), the LF-I frame with fy=650MPa (highest yield stress) fails in a mode exhibiting significant local

deformations, occurring mainly in the vicinity of the beam end support and point of load application

− such local deformations are practically absent from the frame critical buckling mode (see Fig. 10).

5. DSM Design Procedure

The current DSM adopts “Winter-type” design curves, calibrated against experimental and numerical

results concerning the ultimate strengths of single-span columns and beams under uniform compression

and bending (e.g., Schafer 2008). The ultimate strength estimates (Pn for columns and Mn for beams)

against local (Pnl, Mnl), distortional (Pnd, Mnd) and global (Pne, Mne) failures are obtained on the basis of the

(i) elastic buckling loads (Pcrl, Pcrd and Pcre) or moments (Mcrl, Mcrd and Mcre), and (ii) either the cross-

section yield load (Py) and fist-yield moment (My) or the column/beam global strength (Pne or Mne − local/global interactive failure). The expressions providing Pne, Pnl, Pnd, Mne, Mnl and Mnd are given by

( ) yne PP2

ec .0.658λ= if λc.e = crey PP / ≤ 1.5

y2ec

ne PP

=

.

0.877

λ if λc.e > 1.5 , (1)

nenl PP = if λc.l = crlne PP / ≤ 0.776

nene

crl

ne

crlnl P

P

P

P

PP

0.40.4

0.151

−= if λc.l > 0.776 , (2)

ynd PP = if 0.561/. ≤= crdydc PPλ

yy

crd

y

crdnd P

P

P

P

PP

0.60.6

0.251

−= if λc.d > 0.561 , (3)

yne MM = if 600MM creyeb ./. 1.336 , (4)

nenl MM = if 0.776/. ≤= crlnelb MMλ

nene

crl

ne

crlnl M

M

M

M

MM

0.40.4

0.151

−= if 0.776. >lbλ , (5)

ynd MM = if 0.673/. ≤= crdydb MMλ

yy

crd

y

crdnd M

M

M

M

MM

5.05.0

22.01

−= if 0.673. >dbλ , (6)

where (i) λl, λd and λe are the local, distortional and global slenderness values, (ii) Py=Ag.fy and (iii) My=Wy.fy − Ag and Wy are the cross-section area and elastic modulus. In frames, like in other structural systems or in single-span members not subjected to uniform compression

or bending, the various “Pcr and Mcr values” appearing in (1)-(6) (i) cannot adequately describe the loading

and (ii) need to be replaced by “critical buckling load parameter values Qcr”. Moreover, it is worth noting

that the above DSM strength curves neglect both the (i) cross-section elastic-plastic strength reserve and

(ii) bending moment redistribution. Therefore, a more rational approach, already proposed by the

authors in the context of continuous beams (Basaglia and Camotim 2012), consists of replacing

the first-yield (cross-section elastic strength) load (Py) or moment (My) by the “load parameter value

associated with the frame (geometrically linear) plastic collapse” (Qpl) in (1)-(6) − note that, in the case of statically determinate frames, Qpl corresponds to the cross-section plastic strength. In order to be able to assess the quality of the frame ultimate strength estimates provided by the proposed

DSM approach, described in the previous paragraph, the first step consists of determining the Qpl

values for the LF-U, PF-C and LF-I frames, all assumed to exhibit 6 different yield stresses, namely

fy=250, 300, 350, 450, 550, 650 MPa. These Qpl values were obtained through first-order elastic-plastic

1st plastic ringe 2nd plastic ringe

Q=346.16kN Qp=352.41kN

1st plastic

ringe

2nd plastic

ringe

3rd plastic

ringe

Q=85.07kN Qp=85.82kN

1st plastic ringe

2nd plastic ringe

3rd plastic ringe

Q=47.03kN

Q=49.15kN

Qp=52.68kN

LF-U

PF-C

LF-I Figure 14: First-order elastic-plastic deformed configurations and von Misses stress distribution concerning the LF-U, SF-C

and LF-I frames with fy=250MPa and corresponding to their (i) plastic hinge formations and (ii) collapses.

ANSYS SFEA and are given in Table 2. For illustrative purposes, Fig. 14 displays deformed configurations

and von Mises stress distributions, concerning the three frames analyzed and fy=250MPa, which are

associated with (i) the formation of the successive “plastic hinges” (cross-section full yielding) and (ii) the

frame collapse. Note that the members triggering the frame instabilities are again those also responsible

for their (first-order) plastic collapses. Moreover, since the LF-U and PF-C frame members are almost

exclusively subjected to axial compression, the corresponding first plastic hinge locations are not well

defined at all − indeed, full yielding occurs practically at the same time in a whole member “region”. Conversely, the first plastic hinge location is very well defined in the LF-I frame, as it occurs at the beam

end support − in this case, beam non-uniform bending is the primary action (for instability and collapse). Next, the ultimate load estimates yielded by the DSM approach are compared with the ANSYS SFE

values, for the three frames with yield stresses fy=250, 300, 350, 450, 550, 650 MPa. As mentioned

earlier, the current DSM prescribes different strength curve sets for columns (Eqs. (1)-(3)) and beams

(Eqs. (4)-(6)). In this work, the DSM curve set selection was made on the basis of the member triggering

the frame instability and collapse (the same in all cases). This means that DSM column strength curves to

be considered concern (i) columns for the LF-U and PF-C frames, and (ii) beams for the LF-I frames2.

Table 2: Comparison between the beam ultimate load “exact” values and DSM estimates.

SFEA DSM

fy

(MPa) DBN

plQ

(kN)

uQ

(kN) plcr.λ

DSMuQ .

(kN)

*.DSMuQ

(kN) FMN

u

DSMu

Q

Q . u

DSMu

Q

Q*.

250 L 352.41 263.06 1.18 264.95 264.95 L/G 1.01 1.01

300 L 423.12 293.79 1.29 298.01 298.01 L/G 1.01 1.01

350 L 493.03 322.72 1.39 328.47 328.47 L/G 1.02 1.02

450 L 634.64 377.27 1.58 384.86 384.86 L/G 1.02 1.02

550 L 774.28 428.19 1.74 435.16 435.16 L/G 1.02 1.02

LF

-U

650 L 916.78 476.28 1.90 482.29 482.29 L/G 1.01 1.01

250 D 85.82 55.90 1.29 51.62 51.62 D 0.92 0.92

300 D 103.67 61.69 1.42 57.02 57.02 D 0.92 0.92

350 D 121.07 66.81 1.53 61.74 61.74 D 0.92 0.92

450 D 155.89 77.15 1.74 70.00 70.00 D 0.91 0.91

550 D 190.27 85.82 1.92 77.07 77.07 D 0.90 0.90

PF

-C

650 D 224.64 93.40 2.09 83.38 83.38 D 0.89 0.89

250 G 52.68 34.81 1.01 41.84 41.84 G 1.20 1.20

300 G 63.54 38.79 1.11 46.32 46.32 G 1.19 1.19

350 G 73.36 41.10 1.20 49.14 49.14 G 1.20 1.20

450 G 93.39 44.87 1.35 51.32 50.90 L/G 1.14 1.13

550 G 115.89 46.99 1.50 51.32 50.90 L/G 1.09 1.08

LF

-I

650 G 137.10 51.10 1.63 51.32 50.90 L/G 1.01 1.00

2 Note that, since the current DSM does not cover beam-columns, the DSM strength curve selection procedure employed

in this work would be “irreparably compromised” if the instability and collapse of the LF-I frames was triggered by member

A, which is a beam-column − at least, until DSM beam-column strength curves are developed and adequately validated.

Since some of the frames may exhibit “mixed” buckling and failure modes, the DSM estimates obtained

were based on two concepts: (i) failure mode nature assumed to coincide with the “dominant buckling

mode nature” (see Table 1) and slenderness values based on the frame “real” critical buckling load Qcr,

corresponding to a “mixed” buckling mode (not local, distortional or global), and (ii) lower of Qne, Qnl,

Qnd and the three slenderness values based on Qcr.i=(Qb.i /Qb.min) × Qcr, where Qb.min=min {Qb.e, Qb.d, Qb.l}

− usual DSM application. Table 2 provides, for all the frames analyzed, the following values: (i) first-order plastic collapse (Qpl) and ultimate (Qu) loads, (ii) critical slenderness values (λcr.pl), obtained from Qcr (given in section 3) and Qpl, (iii) ultimate load estimates yielded by the current DSM design curves

(iii1) selected on the basis of the dominant buckling nature (Qu.DSM) and (iii2) corresponding to the lower

of Qne, Qnl, Qnd (*

.DSMuQ ). Moreover, this table provides also (i) the dominant buckling natures (DBN),

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5

(Qne /Qcr)0.5

Qne

Qu

DSM local / global curve

for columns and beams

(Qu /Qne)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5

(Qne /Qcr.l)0.5

Qne

Qu

Dominant buckling

mode nature:

Local Global 1

2

3

DSM local / global curve

for columns and beams

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5

(Qpl /Qcr)0.5

Qpl

Qu

(Qu /Qpl)

DSM distortional curve

for columns

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5

(Qpl /Qcr.d)0.5

Qpl

Qu

Dominant buckling

mode nature:

Distortional

DSM distortional curve

for columns

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5

(Qpl /Qcr)0.5

Qpl

Qu (Qu /Qpl)

1

2 3

DSM global curve

for beams

Elastic critical

buckling

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5

(Qpl / Qcr.e)0.5

Qpl

Qu

Dominant buckling

mode nature:

Global

DSM global curve

for beams

Figure 15: DSM ultimate load estimates and SFE Figure 16: DSM ultimate load estimates and “SFE values: “dominant buckling mode nature” values: minimum of Qne, Qnl, Qnd

(ii) the failure mode natures predicted by the DSM (FMN) and (iii) the values of the ratios Qu.DSM /Qu..

On the other hand, Figs. 15 and 16 make it possible to compare the above two sets of DSM estimates

with the values obtained by means of the ANSYS SFE analyses − the small triangles, circles and squares identify the LF-U, PF-C and LF-I frame numerical ultimate strengths, respectively. The observation of the results presented in Table 2 and Figs. 15-16 prompts the following comments and remarks, concerning the “quality” of the DSM-based frame ultimate strength estimates:

(i) The comparison between the SFE ultimate load values and the corresponding DSM estimates based

on the “dominant buckling mode nature”, both of which are displayed in Fig. 15, makes it possible to

conclude that (i1) the LF-U frame predictions (local instability/collapse) are all very accurate, even if

minutely unsafe (Qu.DSM /Qu values varying between 1.01 and 1.02), (i2) the PF-C frame predictions

(distortional instability/collapse) are all safe, even if only reasonable accurate (Qu.DSM /Qu values

varying between 0.89 and 0.92), and (i3) the LF-I frame predictions (global instability/collapse)

are practically all fairly or excessively unsafe (with a single exception, which corresponds to a

“perfect” estimate, the Qu.DSM /Qu values vary between 1.09 and 1.20).

(ii) The adoption of the lower of the Qne, Qnl, Qnd values as the frame ultimate strength estimate only

leads to a change in three LF-I frames, namely those associated with higher slenderness values, i.e.,

yield stresses. They correspond to the points labeled 1-3 in the bottom plot of Fig. 15, which are

found to “migrate” to the bottom plot of Fig. 16, thus meaning that the current DSM application

predicts local/global interactive failures for these three frames. Note also that their slenderness values,

which change from global to local, (ii1) are substantially reduced and (ii2) become identical, which

stems from the fact that Qne=Qcre for λb.e>1.336 and, thus, λb.l=(Qcre/Qcrl)0.5

is constant (see (4)-(5)).

(iii) Since Qcrl is considerably larger than Qcre for the three frames identified in the previous item, the

local/global interaction effects are rather weak, which explains the extreme closeness between the *

.DSMuQ and Qu.DSM values, leading to (minuscule) differences in the corresponding ultimate strength

ratios (see the last three rows in Table 2) − the *.DSMuQ values are a touch below the Qu.DSM ones. So, in this particular case, the DSM global and local/global interactive strength curves predict essentially the

same failure loads, but on the basis of distinct slenderness values.

(iv) The natures of the collapse modes displayed in Figs. 11 (LF-U frame with fy=250 MPa), 12 (PF-C

frame with fy=450 MPa) and 13 (LF-I frame with fy=650 MPa) confirm the DSM predictions.

Indeed, while the first two frames exhibit pure local and distortional collapse modes, respectively, the

third one fails in a mode that combines local and global (flexural-torsional) deformations.

(v) The ultimate strengths of two LF-I frames predicted to fail in local/global interactive modes (those

with critical slenderness values higher than 1.5) are fairly well estimated by the elastic critical

buckling curve3, thus confirming the findings recently reported by the authors (Basaglia & Camotim

2012) in the context of two and three-span continuous beams. Indeed, the elastic critical buckling

curve overestimates these two ultimate strengths by 9% (λcr.pl=1.5) and 1% (λcr.pl=1.63), respectively.

(vi) Due to time limitations, only frames with moderate-to-high critical slenderness values were analyzed.

Results for frames with low, low-to-moderate and high slenderness values will be reported soon. 6. Concluding Remarks

This work reported the results of an ongoing numerical investigation on the local, distortional and global

buckling, post-buckling, collapse and DSM design of simple frames. These results consisted of (i) critical

3 Recall that the current DSM beam global strength curve coincides with the elastic critical buckling curve for λb.e>1.336.

buckling loads and mode shapes, determined through GBT and ANSYS analyses, (ii) post-buckling

equilibrium paths (up to collapse), deformed configurations and von Mises stress distributions, obtained

by means of ANSYS elastic and elastic-plastic SFE analyses, and (iii) ultimate load estimates, provided by

the current DSM strength curves. Out of the findings unveiled in the course of this work, the following

ones deserve to be specially mentioned:

(i) The frame buckling modes often exhibit a “mixed” nature, thus precluding its direct classification

as local, distortional or global. Therefore, it is necessary to resort to the “dominant buckling mode

nature” concept in order to classify the buckling modes− the use of GBT-based buckling analyses makes the application of this concept fairly straightforward.

(ii) Since the (modified) current DSM strength curves were developed and validated in the context of

isolated columns or beams, it was expected that they would only provide satisfactory (safe and

reasonably accurate) ultimate strength estimates in frames that buckle and fail in modes triggered by

members subjected exclusively to pre-buckling axial compression (columns) or bending (beams) − i.e., not beam-columns (members subjected to pre-buckling axial compression and bending).

(iii) Confirming the above assertion, the DSM column design curves predicted fairly well the ultimate

strengths of the LF-U and PF-C frames, which (iii1) buckle in “practically pure” local and distortional

modes, respectively, and (iii2) can be more adequately described as “rigidly connected column sets”

− all its members were subjected exclusively to axial compression. All the local and distortional failure loads were slightly overestimated and underestimated, respectively.

(iv) In the LF-I frames, which (iv1) included one beam and one beam-column, and (iii2) buckled and failed

in predominantly global modes triggered by the beam, the DSM beam strength curves overestimated

the numerical failure loads by an amount that decreased with the frame global slenderness. For

three frames, the lower DSM predictions corresponded to local/global interactive failures (even if the

ultimate strengths obtained were only marginally lower than their global counterparts), which

matched the collapse modes obtained by means of the shell finite element analyses. Finally, it is worth noting that the work reported in this paper is just “the first step of a long journey”,

since a vast amount if research is obviously needed before a direct strength approach for the design of

thin-walled frames can be established on solid grounds. In particular, it is necessary to come up with a

fresh methodology to handle frames whose buckling and failure are triggered by members subjected to

compression and bending, a task directly linked to the development of DSM-based strength curves

and/or methodologies for the design isolated thin-walled beam-columns. Acknowledgments

The authors gratefully acknowledge the financial support of “Fundação para a Ciência e Tecnologia”

(FCT − Portugal), through the research project “Generalised Beam Theory (GBT) − Development, Application and Dissemination” (PTDC/ECM/108146/2008) project. Moreover, the first author is also

indebted to FCT for having granted him the post-doctoral scholarship nº SFRH/BPD/62904/2009. References

American Iron and Steel Institute (AISI) (2007). North American Specification for the Design of Cold-Formed Steel Structural

Members (NAS: AISI-S100-07), Washington (DC).

Basaglia C, Camotim D, Silvestre N (2010). “GBT-Based Buckling Analysis of Thin-Walled Steel Frames with Arbitrary

Loading and Support Conditions”, International Journal of Structural Stability and Dynamics, 10(3), 363-385.

Basaglia C, Camotim D (2012). “Buckling, post-buckling, strength and DSM design of cold-formed steel continuous lipped

channel beams”, Journal of Structural Engineering (ASCE), accepted for publication.

Basaglia C, Camotim D, Silvestre N (2012). “Torsion warping transmission at thin-walled frame joints: kinematics,

modelling and structural response”, Journal of Construction Steel Research, 69, 39-53.

Bijlaard F, Feldmann M, Naumes J, Sedlacek G (2010). “The “general method” for assessing the out-of-plane stability of

structural members and frames and the comparison with alternative rules in EN 1993 – Eurocode 3 – Part 1-1”, Steel

Construction − Design and Research, 3(1), 19-33.

Camotim D, Silvestre N, Basaglia C, Bebiano R (2008). “GBT-based buckling analysis of thin-walled members with non-

standard support conditions”, Thin-Walled Structures, 46(7-9), 800-815.

Camotim D, Basaglia C, Silvestre N (2010). “GBT buckling analysis of thin-walled steel frames: a state-of-the-art report”, Thin-

Walled Structures, 48(10-11), 726-743.

Comité Européen de Normalisation (CEN) (2005). Eurocode 3 − Design of Steel Structures − Part 1-1: General Rules and Rules for Buildings, EN1993-1-1, Brussels.

Dubina D (2008). “Structural analysis and design assisted by testing of cold-formed steel structures”, Thin-Walled Structures,

46(7-9), 741-764.

Hancock GJ, Kwon YB, Bernard ES (1994). “Strength design curves for thin-walled sections undergoing distortional

buckling.” Journal of Constructional Steel Research, 31(2-3), 169-186.

SAS (Swanson Analysis Systems Inc.) (2009). ANSYS Reference Manual (version 12).

Schafer BW, Peköz T (1998). “Direct strength prediction of cold-formed steel members using numerical elastic buckling

solutions”, Proceedings of 14th International Specialty Conference on Cold-Formed Steel Structures (St. Louis, 15-16/10),

R. LaBoube, W.W. Yu (eds.), 69-76.

Schafer BW (2008). “Review: the direct strength method of cold-formed steel member design”, Journal of Constructional

Steel Research, 64(7-8), 766-778.

Standards of Australia (2005). Cold-Formed Steel Structures, Australian Standard/New Zealand Standard 4600

(AS/NZS4600), Sydney.